# Analog of the Birkhoff’s ergodic theorem for the sequence of squares

No – the sequence of squares is universally bad which was proved by Buczolich and Mauldin. I will quote from Tom Ward’s review of their paper Divergent square averages, Ann. of Math. (2) 171 (2010), no. 3, 1479–1530.

A consequence of J. Bourgain’s work [Inst. Hautes Études Sci. Publ. Math. No. 69 (1989), 5–45; MR1019960] is an ergodic theorem along squares, answering earlier questions of Bellow and Furstenberg: If $$(X,mathcal B,T,mu)$$ is a measure-preserving system, then the non-conventional ergodic averages
$$frac1{N} sum_{n=0}^{N-1} f(T^{n^2} x)$$
converge almost everywhere for $$fin L^p$$ with $$p>1$$. Here a comprehensive – and negative – answer is given to his question of whether the result extends to all of $$L^1$$. The authors show that the sequence $$(n^2)$$ is universally bad: for any ergodic measure-preserving system there is a function $$fin L^1$$ for which the above averages fail to converge as $$Ntoinfty$$ for $$x$$ in a set of positive measure.

PS The Birkhoff theorem does not apply to your “particular case” as it requires the presence of a finite invariant measure.

If $$X=mathbb{Z}$$, $$mu$$ is the counting measure, and $$T$$ is the shift operator given by $$Tf(x)=f(x+1)$$, then for all real $$pge1$$, $$finell^p(mathbb{Z})$$, and $$xinmathbb{Z}$$, by Hölder’s inequality,
$$|mathcal{A}_N f(x)|le frac1N,sum_{n=0}^N|f(x+n^2)| lefrac1N,|f|_p,(N+1)^{1-1/p}to0$$
and hence $$mathcal{A}_N f(x)to0$$ as $$Ntoinfty$$.